BRUSSELS — European leaders agreed today to institute a single regulator with broad oversight over banks in the 17-nation euro zone, a step toward binding the countries’ economies more tightly together.

The banking supervisor would have power over the behavior of the roughly 6,000 banks in the euro zone. But the plan would take full effect by the beginning of 2014, later than had been anticipated just weeks ago and on a time frame that may not be quick enough to allay market fears that Europe’s banks and its governments could drag one another down if any of them gets in trouble.

Leaders also discussed Greece’s future in the euro, expressing a statement of support for the country without explicitly saying they would keep it in the currency union.

The banking regulator would delegate supervision of smaller banks to national oversight, a concession to German desires to shield their politically powerful regional banks in an election year and also a concession to the reality that it may be difficult to set up an entirely new regulatory operation over the course of just a few months.

The issue has been contentious, with politicians reluctant to give up national control over their banks and the powerful financial sector worried that European regulators might be less accommodating than local officials, who can be more susceptible to political influence.

But the agreement was in some ways a step back from a June summit in which European officials committed to institute a euro-zone banking supervisor by the end of 2012 without specifying how broad its powers would be.